Egan — Linear Complex, and a certain class of Twisted Curves. 69 



X. — The P-QuINTIC AVITH a BiTANGENX. 



67. Any one of the cuives k (equation 3-i) has the following properties, 

 none of which, as we shall see, belongs to any other B-, : — 



(fl.) It has a bitangent. 



(6) The 2,2 relation breaks up into two symmetrical factors {td = + 1) ; 

 i.e., the secants of the curve which are lines of the complex determine two 

 involutions on the curve. 



(c) There are inscribed in the curve an infinity of quadrilaterals, such 

 that the osculating plane at each vertex contains the two adjacent sides. 

 (The points t, t~\ - t, - t'^ are the vertices of such a quadrilateral.) 



68. We begin with the property (c), and show that in no other rational 

 quintic can oiu such quadrilateral be inscribed. Let A, B, G, D be four 

 nou-coplanar points ; consider a rational quintic passing through them and 

 having BAB, ABG, BOB, GBA as osculating planes at A, B, G, B. Take 

 as z! = 0, t = CO, the foci of the involution on the curve in which A, G and 

 B, B are corresponding pairs. Then if a aud i are the parameters of 

 A and B, those of G and B will be - a and - h. Taking ABGB as tetra- 

 hedron of reference, it is easy to see by considering the points where the 

 coordinate planes meet the curve that the equations of the curve can be 

 written 



li/j iA.'2 w3 'f^\ 



{t + ay {t- -!)■') " (t + bY{f-d') " {t - ay (f - 6') " {t - by {f - a^) ' 



Forming pi^ and ^j^i for the tangent, we find ipn = a^iiu hence the curve 

 belongs to a linear complex. 



The 2, 2 relation works out t-Q'^ = dV-, and if we take ah equal to unity, 

 as we may by multiplying t by l/.y(ab) , we get fO- = 1. 



The line co is a bitangent. For and oo are two ordinary points on the 

 curve, the osculating plane at each of which touches the curve at the other, 

 in virtue of the 2, 2 relation. Property (b) follows from the 2, 2 relation. 

 Hence (a) and (b) are involved in (c). The inflexions are ± 1, ± ^. 



If we now take as coordinate planes the osculating planes at 1, 0, oo, - 1, 

 and consider as before the parameters of the points where these planes meet 

 the curve, we see that the new equations of the curve can be written 



(t -iy{t + 1) ¥ ^ f- ^ {t + iy (i-i)' ^ ' 



In effect, .Tj meets the curve in four points coincident with the inflexion 

 t = 1. It also passes through t = -1, since {t = 1, d = - 1) satisties the 2, 2 

 relation; and so on. 



